Theorem bj-nnst 14498

Description: Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 14745 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( -> , -. ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( -> , <-> , -. )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

Assertion

Ref	Expression
bj-nnst	|- -. -. STAB ph

Proof of Theorem bj-nnst

Step	Hyp	Ref	Expression
1		bj-trst 14494	. 2 |- ( ph -> STAB ph )
2		bj-fast 14496	. 2 |- ( -. ph -> STAB ph )
3	1, 2	bj-imnimnn 14493	1 |- -. -. STAB ph

Syntax hints:   -. wn 3  STAB wstab 830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-stab 831

This theorem is referenced by:  bj-stst  14500  bj-dcst  14516